Question

Assertion :The nature of graph of the functions $f\left(x\right)={\sinh }^2\left(x\right)$ and $g\left(x\right)={\cosh }^2\left(x\right)$ is same. Reason: The two functions are related by the relation $1+{\sinh }^2\left(x\right)={\cosh }^2\left(x\right)$ .

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$f\left(x\right)={\sinh }^{2}x={\left(\frac{e^x-e^{-x}}{2}\right)}^2=\frac{e^{2x}+e^{-2x}-2}{2}--\left(1\right)$

$g\left(x\right)={\cosh }^{2}x=\frac{e^{2x}+e^{-2x}+2}{2}--\left(2\right)$

if $f\left(x\right)-g\left(x\right)=constant$, then $f\left(x\right)$ and $g\left(x\right)$ have same nature of graphs.

Reason: On performing subtraction of $\left(1\right)$ from $\left(2\right)$ we have,

${\sinh }^{2}x-{\cosh }^{2}x=\frac{e^{2x}+e^{-2x}-e^{2x}-e^{-2x}-2-2}{4}=-1$

${\sinh }^{2}x+1={\cosh }^{2}x$